Constraint Propagation Algorithms for Temporal Reasoning
(1986)

This paper considers computational aspects of several temporal representation languages. It investigates an interval-based representation, and a point-based one. Computing the consequences of temporal assertions is shown to be computational/y intractable in the interval-based representation, but not in the poinfbased one. However, a fragment of the interval language can be expressed using the point language and benefits from the tractability of the latter.’ The representation of time has been a recurring concern of Artificial Intelligence researchers. Many representation schemes have been proposed for temporal reasoning; of these, one of the most attractive is James Allen’s algebra of temporal intervals [Allen 831. This representation scheme is particularly appealing for its simplicity and for its ease of implementation with constraint propagation algorithms. Reasoners based on this algebra have been put to use in several ways. For example, the planning system of Allen and Koomen [1983] relies heavily on the temporal algebra to perform reasoning about the ordering of actions. Elegant approaches such as this one may be compromised, however, by computational characteristics of the interval algebra. This paper concerns itself with these computational aspects of Allen’s algebra, and of a simpler algebra of time points. Our perspective here is primarily computation-theoretic. We approach the problem of temporal representation by asking questions of complexity and tractability. In this light, this paper examines Allen’s interval algebra, and the simpler algebra of time points. The bulk of the paper establishes some formal results about the temporal algebras. In brief these results are: l Determining consistency of statements in the interval algebra is NP-hard, as is determining all consequences of these statements. Allen’s polynomial-time constraint propagation algorithm is sound but not complete for these tasks. l In contrast, constraint propagation is sound and complete for computing consistency and consequences of assertions in the time point algebra. It operates in O(n3) time and O(n2) space. l A restricted form of the interval algebra can be formulated in terms of the time point algebra. Constraint propagation is sound and complete for this fragment. Throughout the paper, we consider how these formal results affect practical Artificial Intelligence programs. ‘This research was supported in part by the Defense Advanced Research Agency, under contracts NOOOl4-85-C-0079 and N-0001 4-77-C-0378. Projects The Interval Algebra Allen’s interval algebra has been described in detail in [Allen 831. In brief, the elements of the algebra are relations that may exist between intervals of time. Because the algebra allows for indefiniteness in temporal relations, it admits many possible relations between intervals (213 in fact). But all of these relations can be expressed as vectors of definite simple relations, of which there are only thirteen, 2 The thirteen simple relations, whose definitions appear in Figure 1, precisely characterize the relative starting and ending points of two temporal intervals. If the relation between two intervals is completely defined, then it can be exactly described with a simple relation. Alternatively, vectors of simple relations introduce indefiniteness in the description of how two temporal intervals relate. Vectors are interpreted as the disjunction of their constituent simple relations. A BEFORE B B AFTER A G--v* A MEETS B B MET-BY A A ,B /